The Dirac equation is one of the important equations of relativistic quantum mechanics, quantum electrodynamics, and quantum field theory. The Dirac equation and its quantization in curved spacetime are of great
interest due to the role of spin-\(\frac{1}{2}\)  particles in astrophysics and cosmology. Recent observational confirmation of the expansion of the universe and the quantum field theory demand a detailed investigation of the solutions of the Dirac equation in curved spacetime (see, e.g., [1]}, [2]} and bibliography therein).
The construction of a quantum field theory in curved spacetime relies heavily on a detailed description of the solutions of the Dirac equation in
a curved background. The standard models of Cosmology provide such backgrounds, which form a family of curved backgrounds under the name Friedmann-Lemaître-Robertson-Walker (FLRW) models.
For the FLRW spacetime with accelerated expansion/contraction, more exactly, for the de Sitter spacetime in [3]} a fundamental solution of the Dirac operator and an explicit formula for the solution of the Cauchy problem are obtained. In [4]} an examination of these explicit formulas gave the answer to such an interesting question in the physics of fundamental particles as a validity of the Huygens principle. It must be noted that several exact solutions of the Dirac equation in the FLRW spacetime, which is the basis of the standard cosmological model,
obtained by the separation of variables approach answer to some very
interesting questions of physics (see, e.g., [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]} and bibliography therein). On the other hand, the fundamental solutions to the Dirac operator
can provide information that is impossible to obtain by numerical solutions of differential equations, as well as new solutions that are not among the exact solutions obtained by the separation of variables approach.
